The area of a rectangle at point *x* is given by:

A=xe^{-x}

The maximum (or minimum) will occur when

`(dA)/(dx)=0`

Now

`(dA)/(dx)=(x)(-e^(-x))+(e^(-x))(1)`

`=e^(-x)(1-x)`

This expression only equals zero when *x* = 1, and is positive when *x* < 1 and negative when
*x* > 1, so we have a maximum.

So the maximum area is (1)(e^{-1}) =
0.3679 units^{2}